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Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry. (English) Zbl 1191.39011
Authors’ abstract: We study the existence of stationary solutions for the following discrete vector nonlinear Schrödinger equation
\[ i\frac{\partial\varphi_n}{\partial t}= -\Delta\varphi_n+ \tau_n\varphi_n- Jf(n,|\varphi_n|)\varphi_n, \] where \(\varphi_n\) is a sequence of 2-component vector, \(J=\left(\begin{smallmatrix}0&1\𝟙&0\end{smallmatrix}\right)\), \(\Delta\varphi_n= \varphi_{n+1}+ \varphi_{n-1}- 2\varphi_n\) is the discrete Laplacian in one spatial dimension and sequence \(\tau_n\) is assumed to be \(N\)-periodic in \(n\), i.e. \(\tau_{n+N}=\tau_n\). We prove the existence of infinitely many nontrivial stationary solutions for this system by variational methods. The same method can also be applied to obtain infinitely many breather solutions for single discrete nonlinear Schrödinger equation.

MSC:
39A12 Discrete version of topics in analysis
35Q55 NLS equations (nonlinear Schrödinger equations)
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